On the Hilbert 2-Class Field Tower of Some Imaginary Biquadratic Number Fields

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{k} = \mathbb{Q} \left( {\sqrt 2 ,\; \sqrt d } \right)$$\end{document} be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{k}_2^{\left( 2 \right)}$$\end{document} its second Hilbert 2-class field. Denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = {\rm{Gal}}\left( {\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}} \right)$$\end{document} the Galois group of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}}$$\end{document}. The purpose of this note is to investigate the Hilbert 2-class field tower of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{k}$$\end{document} and then deduce the structure of G.